Method for interpreting yaw data in a projectile traversing a resisting medium

ABSTRACT

Snapshot data from a projectile traversing a resisting medium are analyzedo determine a lower critical value within the resisting medium where rapid yaw growth begins and a yaw-growth constant governing yaw growth from the lower critical value to an upper critical value. Snapshot data from additional projectiles are similarly analyzed. Curves produced by the solution of a differential equation, including the lower critical value and the yaw-growth constant, are overlaid graphically or analytically to provide an increased data density sufficient to improve substantially the ability to predict the yaw performance of the projectile.

GOVERNMENTAL INTEREST

The invention described herein may be manufactured, used and licensed byor for the Government for Governmental purposes without the payment tome of any royalties thereon.

BACKGROUND OF THE INVENTION

The present invention relates to techniques for interpreting data frommeasurements in which the data density available from a singlemeasurement is too sparse to provide an understanding of the underlyingprocess being measured.

Measurements of high-speed ballistic projectiles, traversing a resistingmedium are conventionally performed using high-speed optical and X-raycameras taking snapshots of the projectile at a few discrete points inits travel. In practice, about four or five snapshots can be obtainedfor a single traversal.

The manner in which the yaw of a projectile grows during traversal is ofinterest to weapon designers. It is desired to interpret snapshotmeasurement data of a traversing projectile in a way permitting anunderstanding of yaw growth with sufficient detail to permit analysis offactors affecting it and for enabling mathematical modeling of the yawgrowth. The sparsity of data points measurable from a single projectiledoes not enable the desired understanding. An attempt to average thedata points from several projectiles at corresponding points within themedium fails to yield a useful result since the position within themedium where significant yaw growth begins is subject to many variablesoutside the control of the experimenter.

OBJECTS AND SUMMARY OF THE INVENTION

Accordingly, it is an object of the invention to provide adata-interpretation technique which solves the problems of the priorart.

More particularly, it is an object of the invention to provide adata-interpretation technique for a projectile traversing a resistingmedium which permits the development of a density of measurement pointssufficient to foster an understanding of the underlying process.

It is a further object of the invention to provide a techniquepermitting the combination of snapshot measurements from separateprojectile traversals of a resisting medium to provide a consistent setof data points having a density sufficient for supporting anunderstanding of the principles of yaw growth.

Briefly stated, the present invention provides a technique for analyzingsnapshot data from a projectile traversing a resisting medium. Thesparse data points are analyzed to determine a lower critical valuewithin the resisting medium where rapid yaw growth begins and ayaw-growth constant governing yaw growth from the lower critical valueto an upper critical value. Snapshot data from additional projectilesare similarly analyzed. Curves produced by the solution of adifferential equation, including the lower critical value and theyaw-growth constant, are overlaid graphically or analytically to providean increased data density sufficient to improve substantially theability to predict the yaw performance of the projectile.

According to an embodiment of the invention, there is provided a methodfor interpreting yaw data in a projectile traversing a medium comprisingthe steps of: firing a first projectile through the medium, taking afirst plurality of measurements of position and yaw angle of the firstprojectile during its travel through the medium, solving an equation forthe yaw angle using at least some of the first plurality of measurementsto derive a first critical lower value of the yaw angle and a first yawangle constant for the first projectile, firing a second projectilethrough the medium, taking a second plurality of measurements ofposition and yaw angle of the second projectile during its travelthrough the medium, solving the equation for the yaw angle using atleast some of the second plurality of measurements to derive a secondcritical lower value of the yaw angle and a second yaw angle constantfor the second projectile, and overlaying the second critical lowervalue on the first critical lower value to produce a composite curvecontaining the first and second pluralities of measurements.

The above, and other objects, features and advantages of the presentinvention will become apparent from the following description read inconjunction with the accompanying drawings, in which like referencenumerals designate the same elements.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a schematic diagram showing several stages in the penetrationof a resisting medium by a projectile.

FIG. 2 is an enlarged view of a projectile of FIG. 1.

FIG. 3 is a set of curves drawn between data points of three sets ofmeasurements of yaw of separate projectiles traversing a resistingmedium.

FIGS. 4A, 4B and 4C are curves derived from the three sets ofmeasurements of FIG. 3 by the solutions of an equation based on the datapoints.

FIG. 5 is a composite curve produced by overlaying lower critical valuesin the curves of FIGS. 4A, 4B and 4C to produce a dense data set.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT

Referring to FIG. 1, a resisting medium 10 is interposed in the path ofa projectile 12 which may be, for example, a bullet. An originaldirection of flight of a center of mass 14 of projectile 12 before itenters resisting medium 10 is indicated by a z axis 16. An r axis 18 isdefined normal to z axis 16.

Resisting medium 10 includes a penetration surface 20 through whichprojectile 12 enters. Resisting medium 10 may be any convenient materialsuch as, for example, water, loose earth or hydrocarbon materials. Inthe preferred embodiment, resisting medium 10 is a block of transparentgelatin material chosen because of its uniform resistance to the passageof projectile 12 and for its light transparency which permits takingoptical photographs. In addition, since a block of gelatin material canbe self supporting, resisting medium 10 needs no container. Whennon-self-supporting materials are employed in resisting medium 10, asuitable container is required.

Referring now to FIG. 2, longitudinal axis of projectile 12, representedby an x axis 22, is inclined by a yaw angle a 24 to the instantaneousvelocity vector represented by a velocity vector axis V 26. A y axis 28is defined normal to x axis 22.

Prior to reaching a point of penetration 30 on penetration surface 20,velocity vector axis V 26 coincides with z axis 16. At penetration, aninitial yaw angle a 24A (FIG. 1) exists due to precession, cross-trackvelocity or wobble developed in the external path of projectile 12. Inaddition, the plane of penetration surface 20 may depart from normal toz axis 16 and velocity vector axis V 26 at the moment of entry. All ofthese factors, which cannot be explicitly measured or predicted for aparticular penetration, have a profound effect on the desired yawmeasurements.

During an initial range of travel to a position indicated by the suffixB, center of mass 14 of projectile 12 generally follows z axis 16 with aslowly growing yaw angle a 24. At some critical value, the slow growthof yaw angle a 24 changes to a rapid growth such as indicated atsuccessive positions of center of mass 14 and projectile 12 indicated bythe suffixes C and D. Projectile 12 develops lift which bends the pathof projectile 12 along a curved path whose tangent represents the localdirection of velocity vector axis V 26. Yaw angle a 24 may grow toexceed 90 degrees as indicated at positions of center of mass 14 andprojecticle 12 indicated by the suffixes E and F.

I have discovered that a yaw angle a 24 of 90 degrees is a stablecondition and that growth of yaw angle a 24 slows or even reverses afterreaching a maximum value. The greatly increased drag as yaw angle a 24approaches 90 degrees may prevent growth of yaw angle a 24 to itsmaximum value. I have observed that a reversal in the growth of yawangle a 24 sometimes does occur and such occurrence complicates theinterpretation of the measured data.

Referring now to FIG. 3, typical data points of yaw versus penetrationdistance as measured by high-speed photography for three projectileswith the points joined by lines. It will be noted that the curves failto overlie each other. I have discovered that the conventional techniqueof averaging the yaw values at selected values of penetration does notyield a result leading to understanding the yaw phenomenon, nor provideguidance for developing a mathematical model.

Referring now to FIGS. 4A, 4B and 4C in which the three sets of datapoints from FIG. 3 are separately plotted, I have discovered that, if asmooth curve is fitted to the data points using only yaw values fallingbetween a lower critical value and an upper critical value, theresulting smooth curves have an almost identical shape, but aredisplaced in the horizontal direction. The smooth curve is calculated bysolving the following differential equation for M and s_(o) :

    d.sup.2 a/d s.sup.2 =M sin a cos a

Where:

M=yaw growth constant

s_(o) =position where lower critical value of yaw occurs and where theother initial condition, da/ds at s_(o), can be obtained from M ands_(o) due to an additional restriction on the differential equation.

The position along the path where rapid yaw growth begins is criticallydependent upon the conditions existing as projectile 12 (FIGS. 1 and 2)penetrates penetration surface 20. Since these conditions are notavailable, the position along z axis 16 at which rapid yaw growth beginsmay vary widely. This appears to account for the fact that the measuredyaw at a particular penetration for one projectile varies substantiallyfrom that measured for another projectile.

I have discovered that, if each smooth curve is extrapolatedhorizontally to a lower critical value of penetration at which rapid yawgrowth begins and, if that value is identified as the origin of acomposite curve, then all of the curves can be overlaid one upon theother in a manner which finds virtually all of the data points on asingle curve. Such a composite curve is illustrated in solid line inFIG. 5 wherein the abscissa s is the penetration beyond the criticalvalue, in calibers.

The value chosen as the defining lower critical value for the onset ofrapid yaw growth may vary with different resisting medium 10 andprojectile 12. In the preferred embodiment, the lower critical value isfrom about 3 to about 10 degrees and, in the most preferred embodiment,the lower critical value is from about 5 to about 7 degrees. The mostpreferred lower critical value is about 6 degrees.

Beyond the upper critical value, projectile 12 is advancing baseforward. Yaw growth constant M no longer applies. In the upper range liedata points in a region wherein yaw growth is slowing or reversing.Attempts to account for these points previously distorted the measureddata. With my invention, these points are also found to fall on a smoothcurve. Since the present invention permits the superposition of datafrom a large number of projectile penetrations, the density of datapoints can be made as great as desired to indicate the true manner inwhich yaw varies during its travel. With sufficient data points, thevariation of yaw angle at large penetrations can be traced as indicatedby a dashed line.

Prior-art methods are incapable of predicting yaw growth beyond yawangles of about 20 degrees whereas the present invention is surprisinglyaccurate up to yaw angles of as great as 140 degrees.

Having described preferred embodiments ofthe invention with reference tothe accompanying drawings, it is to be understood that the invention isnot limited to those precise embodiments, and that various changes andmodifications may be effected therein by one skilled in the art withoutdeparting from the scope or spirit of the invention.

What is claimed is:
 1. A method for interpreting yaw data in aprojectile traversing a medium comprising the steps of:firing a firstprojectile through said medium; taking a first plurality of measurementsof position and yaw angle of said first projectile during its travelthrough said medium; solving an equation for said yaw angle using atleast some of said first plurality of measurements to derive a firstcritical lower value of said yaw angle and a first yaw angle constantfor said first projectile; firing a second projectile through saidmedium; taking a second plurality of measurements of position and yawangle of said second projectile during its travel through said medium;solving said equation for said yaw angle using at least some of saidsecond plurality of measurements to derive a second critical lower valueof said yaw angle and a second yaw angle constant for said secondprojectile; and overlaying said second critical lower value on saidfirst critical lower value to produce a composite curve containing saidfirst and second pluralities of measurements.
 2. A method according toclaim 1 wherein said first and second critical lower values are equaland from about 3 to about 10 degrees.
 3. A method according to claim 2wherein said first and second critical lower values are equal and fromabout 5 to about 7 degrees.
 4. A method according to claim 1 whereinsaid at least some of said first plurality include only yaw values lowerthan a first upper critical value and said at least some of said secondplurality include only yaw values lower than a second upper criticalvalue.
 5. A method according to claim 4 wherein said first and secondupper critical values are equal.
 6. A method according to claim 4wherein said first and second upper critical values are above values atwhich said first and second yaw angle constants are valid.
 7. A methodaccording to claim 1 wherein the step of overlaying includes graphicallyoverlaying first and second curves.
 8. A method according to claim 1wherein said step of overlaying includes analytically overlaying firstand second equations representing curves.
 9. A method according to claim1 wherein said equation has the form:

    d.sup.2 a/d s.sup.2 =M sin a cos a

Where: M=yaw growth constant s_(o) =position where lower critical valueof yaw occurs and where the other initial condition, da/ds at s_(o), canbe obtained from M and s_(o) due to an additional restriction on thedifferential equation.